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Methods of integration for OCR A-level Maths

Methods of integration

This page covers the following topics:

1. Integration by parts
2. Integration by substitution
3. Integrating partial fractions

Integration by parts is a special method of integration which is used when two functions are being multiplied. They are defined as u and v accordingly, and the given formula is used to evaluate the interval.

Integration by parts

Integration by substitution is a method to evaluate integrals by changing the variables. This method can be used when the integral is of the given form.

Integration by substitution

When the integrand is in the form of a proper fraction, the integral should be rewritten as partial fractions and then integrated using the given result.

Integrating partial fractions

1

Evaluate โˆซ1/(x + 5)dx.

โˆซ1/(x + 5)dx = ln|x + 5| + c, where c is the constant of integration.

Evaluate โˆซ1/(x + 5)dx.

2

Use integration by parts to find โˆซ(5 + x)cos(x)dx.

Let u = 5 + x and dv = cos(x)dx.
Then, du = dx and v = sin(x).
โˆซ (5 + x)cos(x)dx = (5 + x)sin(x) โˆ’ โˆซsin(x)dx =
= (5 + x)sin(x) + cos(x) + c, where c is the constant of integration

(5 + x)sin(x) + cos(x) + c

Use integration by parts to find โˆซ(5 + x)cos(x)dx.

3

Determine the following integral: โˆซ3xยฒsin(xยณ โˆ’ 10x) โˆ’ 10sin(xยณ โˆ’ 10x)dx.

The integral can be rewritten as โˆซ(3xยฒ โˆ’ 10)sin(xยณ โˆ’ 10x)dx. To integrate by substitution, let u = xยณ โˆ’ 10x and du = (3xยฒ โˆ’ 10)dx. So, โˆซ(3xยฒ โˆ’ 10)sin(xยณ โˆ’ 10x)dx = โˆซsin(u)du = โˆ’cos(u) + c = โˆ’cos(xยณ โˆ’ 10x) + c, where c is the constant of integration.

Determine the following integral: โˆซ3xยฒsin(xยณ โˆ’ 10x) โˆ’ 10sin(xยณ โˆ’ 10x)dx.

4

Evaluate โˆซxeหฃ.

Let u = x and dv = eหฃdx. Then, du = 1dx and v = eหฃ. So, โˆซxeหฃ = xeหฃ โˆ’ โˆซeหฃdx = xeหฃ โˆ’ eหฃ + c, where c is the constant of integration.

Evaluate โˆซxeหฃ.

5

Integrate the area of the given rectangle.

The integral is โˆซ15xยฒsin(8 โˆ’ 5xยณ)dx. To integrate by substitution, let u = 8 โˆ’ 5xยณ and du = โˆ’15xยฒdu. So, โˆซ15xยฒsin(8 โˆ’ 5xยณ)dx = โˆซโˆ’sin(u)dx = cos(u) + c = cos(8 โˆ’ 5xยณ) + c, where c is the constant of integration.

Integrate the area of the given rectangle.

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